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For potential energy of two-layer entrainment with constant $B=g^{\prime }h$, the fixed-energy turbulent mixed layer assumption gives $u_U^{2}h=C_K^2{\Omega }^2{R}^2{h}_0$. If $u_I=C_I{u}_U$ and the interfacial power model for entrainment applies with fixed coefficients, integration gives the displayed depth law with $A=5C_W{c}_D{C}_I^3{C}_K^3{\Omega }^2{R}^3/(B{h}_0)$. The relation applies only before the layer reaches the tank bottom. Its interfacial Richardson number $B{d}_I/(h{u}_I^2)$ stays constant for fixed interface thickness, whereas it decreases like $h^{-1}$ in the constant-speed entrainment model.

An interfacial stress $\tau =c_D{\rho }_l{u}_I^2$ supplies local stress power $P_I=S\tau u_I$. Assuming a fixed fraction $C_W$ becomes global potential energy gives ${\dot{E}}_P=C_W{P}_I$. Together with potential energy of two-layer entrainment, this gives $\dot{h}=2C_W{c}_D{u}_I^3/(g^{\prime }h)$. This is an additional local power-transfer closure. The shaft power of a rotating lid is its torque times its prescribed angular speed and need not equal $P_I$. Geometric factors and a constant transfer fraction may be absorbed into the empirical coefficients only when explicitly assumed time-independent.

Let $h_0=h(0)$ and $S=\pi R^2$. The closed tank has constant mass, soTherefore $({\rho }_l-{\rho }_u)h=({\rho }_l-{\rho }_u(0))h_0$. Under the Boussinesq approximation, define $g^{\prime }=g({\rho }_l-{\rho }_u)/{\rho }_l$ and obtainThe gravitational potential energy is especially easy to evaluate with the stipulated downward depth coordinate: the gravitational potential per unit mass is $-gz$. HenceThis proves the potential energy of two-layer entrainment relationThe positive sign expresses the lifting and homogenization of dense entrained fluid within the upper layer. The fixed thin interface is neglected in this leading two-layer energy integral.

To connect the prescribed interfacial drag coefficient to the energy supply, state the lumped interfacial power model for entrainment explicitly:It estimates the power transmitted by lid-driven flow to the mixing interface. Constant geometric and transfer factors are included in $c_D$ or $C_W$. It is the local stress-power interpretation required for the depth laws in this question, not an identity equating arbitrary lid shaft power to interfacial power. With the potential energy of two-layer entrainment from (a),In model S, $u_I=\Omega R$ and $B$ are constant, soThus the depth increases at a constant rate proportional to the inverse bulk Richardson number, with $h=h_0+2C_W{c}_D\Omega Rt/{\mathrm{Ri}}_B$, until the layer reaches $H$. Here the prescribed $u_I$ must be read as a characteristic speed. Literal pure solid-body rotation has zero radial velocity, so the PDF's parenthetical description of $u_I$ as radial cannot simultaneously be a strict component statement and the assumption $u_I=\Omega R$.

If the phrase about lid work is taken literally as shaft power $P_{\mathrm{lid}}=\Omega {\mathcal{T}}_{\mathrm{lid}}$, the interfacial stress alone does not specify the lid torque or its energy transfer to the interface. The preceding closure is therefore a visible model assumption, not a hidden consequence of solid-body rotation.